The theory of laser absorption spectroscopy is well understood and has been published previously [
33,
34,
35,
36]. The principles and techniques of the LCF-WMS method are briefly reviewed in
Section 2.1 to define terms and guide the discussion. The selection of transition lines for three components is introduced in
Section 2.2. The strategy of NO and NO
2 detection under high H
2O interference is shown in
Section 2.3.
2.1. Fundamentals of Linear Calibration-Free Wavelength Modulation Spectroscopy
When the collimating laser beam passes through a uniform gas medium, the spectral transmissivity τν�� is defined according to the ratio between the transmitted beam intensity It�� and the incident beam intensity I0�0, which is described by the Beer–Lambert law:
where αν�� is the spectral absorbance at frequency ν, L (cm) is the optical path length, and kν�� (cm−1) is the spectral absorption coefficient, which is given by
where P (atm) is the gas pressure, X is the mole fraction of the absorbing gas species, S(T) (cm−2 atm−1) is the temperature-dependent line strength, and ϕν�� is the line shape function. The line shape function ϕν�� (cm) can be normalized, and therefore the integral of ϕν�� over the entire frequency range is equal to one.
The wavelength of the laser is tuned using a combination of a high-frequency sinusoidal modulation and a lower-frequency scan signal. The frequency modulation and intensity modulation can be expressed as
where ν¯(t)�¯(�) (cm−1) is the center frequency of the slower wavelength scan signal while the laser is modulated at frequency f�, a(t)�(�) (cm−1) is the modulation depth, and ψ� is the frequency modulation phase; I¯0(t)�¯0(�) is the incident laser intensity without modulation, im(t)��(�) is the m-th Fourier coefficient of the laser intensity, and ψm�� is the phase of the m-th laser intensity modulation.
The logarithm of the transmitted laser intensity can be written as
After the logarithmic calculation, the transmitted light intensity consists of three parts: the optical-electronic gain intensity term, the intensity modulation term, and the absorption signal term. The harmonic signal is extracted from ln(It(t))ln(��(�)) by lock-in amplifier, and the k-th absorption harmonic signal becomes
After subtracting the background, harmonic signals are theoretically independent of the laser intensity characteristics. The measured harmonics are only related to the time–frequency relationship (the line shape derivative dk+2gϕvdνk+2g��+2������+2� and the modulation depth a(t)�(�)) and the integrated absorbance. During the experiments, the time–frequency relationship can be obtained using a Fabry–Perot interferometer. The signals detected are only associated with the spectral features integrated along the LOS.
Owing to the decoupling of the laser characteristics and the integrated absorbance, the influence of gas properties can be clearly reflected in the harmonics. This approach allows the numerical simulation of large absorption harmonics without considering the effect of laser intensity modulation characteristics. Therefore, the LCF-WMS approach is suitable for trace gas detection in spite of the strong, broad spectral H2O interference in non-uniform fields for the following reasons: compared with the DAS approach, it eliminates the dependence on the baseline and suppresses the low-frequency noises; compared with the normalized WMS approach, it decouples the characteristics of the light source and the flow field, making the measured signals related to the spectral features integrated along the line of sight (LOS).
2.2. Wavelength Selection
The selection of absorption transitions is critical from the viewpoint of multi-wavelength sensor design. Considering the composition of typical combustion products, H
2O takes a large proportion (~10%), while NO
x molecules are trace components (~ppm). For the detection of NO and NO
2 generated from combustion, spectroscopic signals can be affected by the blended neighboring features and interfering species spectrum, which decreases the SNR and influences the measurement accuracy. These require transition lines with stronger absorption and weaker interference from the other combustion species [
37,
38,
39]. To ensure a sufficient SNR of the absorption signal, the peak absorbance should be above 0.001.
shows a broadband spectral simulation of typical components in near- and mid-infrared bands at a representative exhaust temperature of 600 K based on the HITRAN database [
40]. The fundamental vibration band of NO near 5.2 µm and NO
2 near 6.25 µm holds the most promising candidates, which has the strongest absorption band and only primary interference from H
2O. In this circumstance, it is important to acquire an accurate H
2O concentration for the following spectral resolution. The overtone and combination bands within 1.3–1.5 µm were investigated for H
2O sensing because of the maturity of near-infrared DFB lasers.
Figure 1. Absorption line strength for near- and mid-infrared bands of typical combustion exhaust at 600 K based on HITRAN 2020 [
40]. Transitions with a line strength of less than 5 × 10
−24 cm/molecule are not shown.
According to the aforementioned criteria, the absorption lines in the fundamental band for NO and NO2, as well as the overtone and combination band of H2O, were investigated. For H2O, several candidates (7456.1 cm−1, 7457 cm−1, and 7644.6 cm−1) are plotted in a. The absorption line at 7644.6 cm−1 was finally selected for the following reasons: (1) 7644.6 cm−1 is well isolated; (2) this transition has negligible interference from other absorption species; (3) compared with the strongest absorption line, 7456.1 cm−1, this transition has a smaller variation in the line strength within the temperature range of 400–700 K (shown in b), which is less sensitive to temperature changes and more suitable for non-uniform field measurements. The validation of the line strength of the selected H2O transitions was carried out in an ambient environment. The deviation between the measured line strength and the HITRAN database was less than 8%, while the uncertainty of the spectral parameter was in the range of 5–10%.
Figure 2. Absorbance simulation of three candidate H2O transitions (a) and the corresponding line strength varying from 300 to 900 K (b). P = 1 atm, L = 2 m, T = 600 K, XH2O = 10%, XCH4 = 0.2%, XCO2 = 3%.
The promising transitions near 1909.13 cm
−1 and 1599.9 cm
−1 were selected for NO and NO
2, respectively. The line strength of the selected NO lines was validated in [
41], and it agrees with the uncertainty provided by the HITEMP database (5–10% uncertainty) [
42]. For NO
2 candidates near 1599.9 cm
−1, a reliable collisional broadening parameter database was established by Sur et al. [
28]. The simulated spectra of typical components near the target lines are plotted in , which indicates that the H
2O absorptions have major interference from those of NO and NO
2. In both spectrum regions, the absorption features of H
2O are smooth and weak at the intermediate temperature. However, several new water spectral features appear with the temperature elevation, making the problem more complicated. In such circumstances, the interference from weak, high-internal-energy water vapor transitions cannot be eliminated by wavelength selection. The strategy for the spectral resolution is discussed in the next section. It must be mentioned that CH
4 has negative influences on the detection of NO
2. The influence of CH
4 becomes significant along with the temperature increases.
Figure 3. Simulated absorption spectra of the selected H2O, NO, and NO2 transitions near 1.31, 5.2, and 6.25 µm for typical combustion exhaust: XH2O = 10%; XNO = 50 ppm; XNO2 = 10 ppm; XCH4 = 0.2%; XCO2 = 3%; P = 1 atm; L = 2 m; T = 300, 600, and 900 K.